It can't be "more than two years old," since it the 6th gen iPod nano was released a littile less than two years ago. But unless you bought AppleCare with it (and it's more than ONE year old), I guess it is still out of warranty...
That "diagnostic mode N20" is no "weird" mystery. It can be accessed using this procedure
Doing a Reset
should (normally) restart the iPod and take you out of this mode. You've probably done the Reset, but did you (or the Apple Store person) do a Restore of the iPod in iTunes. Select iPod in iTunes sidebar, under DEVICES. Over to the right, go to the Summary tab. There is a Restore button there. This will erase the iPod, re-install its software, and set it to default settings. If a software glitch is causing this problem, it should be fixed.
What does the error say, when you do a Restore?
The fact that you can DO a Restore in iTunes means that when you connect the iPod to your computer, it acts normally (no more "diagnostic mode N20" screen), correct? If it was still in "diagnostic mode N20," it would not appear in iTunes, so you would not be able to even try doing a Restore.
So, when you are in that position to try doing a Restore, what happens when you eject and disconnect the iPod? Does the iPod just restart and go back to "diagnostic mode N20"? That seems unlikely...